Yes, many of the indoor, sit-down places catering to expats will let you in before prayer and you can order your meal and continue as normal. Some will even let you in during prayer if you just knock and look confused ;-)

Lots of good places, for locals and expats. Mondo's for Italian-ish, Sanbok for good (expensive) seafood, Abu Nawas is a bit of a touristy Syrian-esque outfit but cheap and good. Mandi King for good Arabic style kebsa (cheap). For good Middle Eastern fast food, try Shellal (several locations) or any number of the chawarma joints in the Doha area of town (don't know the name of my favorite, just the location).

As with many places outside the US and Europe, great food is on the whole very affordable.

I only go to the chain restaurants when dragged in a group. Most taste pretty much the same and prices are comparable.